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Properties

Label 2-270-135.113-c1-0-16
Degree $2$
Conductor $270$
Sign $0.497 + 0.867i$
Analytic cond. $2.15596$
Root an. cond. $1.46831$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.996 + 0.0871i)2-s + (0.319 − 1.70i)3-s + (0.984 + 0.173i)4-s + (2.02 − 0.946i)5-s + (0.466 − 1.66i)6-s + (−2.76 − 1.93i)7-s + (0.965 + 0.258i)8-s + (−2.79 − 1.08i)9-s + (2.10 − 0.766i)10-s + (0.0536 − 0.147i)11-s + (0.609 − 1.62i)12-s + (0.614 + 7.02i)13-s + (−2.58 − 2.16i)14-s + (−0.964 − 3.75i)15-s + (0.939 + 0.342i)16-s + (−2.66 + 0.713i)17-s + ⋯
L(s)  = 1  + (0.704 + 0.0616i)2-s + (0.184 − 0.982i)3-s + (0.492 + 0.0868i)4-s + (0.906 − 0.423i)5-s + (0.190 − 0.681i)6-s + (−1.04 − 0.731i)7-s + (0.341 + 0.0915i)8-s + (−0.932 − 0.362i)9-s + (0.664 − 0.242i)10-s + (0.0161 − 0.0444i)11-s + (0.176 − 0.467i)12-s + (0.170 + 1.94i)13-s + (−0.691 − 0.579i)14-s + (−0.249 − 0.968i)15-s + (0.234 + 0.0855i)16-s + (−0.645 + 0.173i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.497 + 0.867i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.497 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(270\)    =    \(2 \cdot 3^{3} \cdot 5\)
Sign: $0.497 + 0.867i$
Analytic conductor: \(2.15596\)
Root analytic conductor: \(1.46831\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{270} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 270,\ (\ :1/2),\ 0.497 + 0.867i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.75838 - 1.01896i\)
\(L(\frac12)\) \(\approx\) \(1.75838 - 1.01896i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.996 - 0.0871i)T \)
3 \( 1 + (-0.319 + 1.70i)T \)
5 \( 1 + (-2.02 + 0.946i)T \)
good7 \( 1 + (2.76 + 1.93i)T + (2.39 + 6.57i)T^{2} \)
11 \( 1 + (-0.0536 + 0.147i)T + (-8.42 - 7.07i)T^{2} \)
13 \( 1 + (-0.614 - 7.02i)T + (-12.8 + 2.25i)T^{2} \)
17 \( 1 + (2.66 - 0.713i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-4.34 + 2.51i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.32 - 6.16i)T + (-7.86 + 21.6i)T^{2} \)
29 \( 1 + (0.210 - 0.176i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-1.75 + 9.96i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (-0.581 - 2.17i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (4.27 - 5.09i)T + (-7.11 - 40.3i)T^{2} \)
43 \( 1 + (-1.49 - 3.20i)T + (-27.6 + 32.9i)T^{2} \)
47 \( 1 + (1.25 - 1.79i)T + (-16.0 - 44.1i)T^{2} \)
53 \( 1 + (4.14 - 4.14i)T - 53iT^{2} \)
59 \( 1 + (8.92 - 3.24i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-1.06 - 6.05i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (5.61 - 0.491i)T + (65.9 - 11.6i)T^{2} \)
71 \( 1 + (6.34 + 3.66i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.43 + 12.8i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (4.50 + 5.36i)T + (-13.7 + 77.7i)T^{2} \)
83 \( 1 + (-0.118 + 1.35i)T + (-81.7 - 14.4i)T^{2} \)
89 \( 1 + (2.96 + 5.13i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.50 - 1.16i)T + (62.3 - 74.3i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.92200028586677143894371858952, −11.18890503395882393712614611431, −9.619699001472289945239575994347, −9.070624122595025936755398279523, −7.45801555119243507894589995154, −6.64852056237281700388856673616, −5.98455442656317105216649480255, −4.49538102123462415253143122725, −3.01876578395202675109467455150, −1.56068199552837379676389076918, 2.75376611193207044247601633690, 3.30247409826628512084532513782, 5.11233746904946897602022074971, 5.73655665648451057011667000090, 6.78476553745611526788408536372, 8.440341178798023251686435467367, 9.460166298233095636080662316717, 10.31201605467124510941214370210, 10.85152279912382154303270975639, 12.30240166852566908651773090436

Graph of the $Z$-function along the critical line